On the determination of series, or a new method for finding the general terms of series
This paper attempts to determine f(x), given f(1), f(2), f(3), etc. Euler begins with the example f(n) = n and tries f(x) = x + b1∙sin(Ï x) + b2∙sin(2Ï x) + b3∙sin(3Ï x) + .... Note that f(x) – x must be periodic, with f(x+1) = f(x). Then he gets f(x+1) by Taylor series at f(x) and obtains a differential equation of infinite order. A few paragraphs later, we have a perfect Fourier series.
Original Source Citation
Novi Commentarii academiae scientiarum Petropolitanae, Volume 3, pp. 36-85.
Opera Omnia Citation
Series 1, Volume 14, pp.463-515.